Another Approach to Solution. of Fuzzy Differential Equations

Transcription

1 Applied Memil Siees, Vol. 4, 00, o. 6, Aoer Appro o Soluio o Fuzz Diereil Equios C. Durism Deprme o Memis ogu Egieerig College Peruduri, Erode Tmildu, Idi d@kogu..i B. Us Deprme o Memis ogu Egieerig College Peruduri, Erode Tmildu, Idi viks_us@oo.o.i Absr I is pper, we ve irodued d sudied ew eique or geig e soluio o uzz iiil vlue problem. Memis Subje Clssiiio: 4A; 65L05 ewords: Fuzz diereil equios, Ruge-u meod o order ree, Trpezoidl uzz umber Iroduio Fuzz diereil equios re url w o model dmil ssems uder ueri. Firs order lier uzz diereil equios re oe o e simples uzz diereil equios, wi pper i m ppliios.

2 778 C. Durism d B. Us I e ree ers, e opi o FDEs s bee ivesiged exesivel. Te oep o uzz derivive ws irs irodued b S. L. Cg d L. A. Zde i []. I is pper, we ve irodued d sudied ew eique or geig e soluio o uzz iiil vlue problem. Te orgized pper is s ollows: I e irs ree seios, we rell some oeps d iroduor merils o del wi e uzz iiil vlue problem. I seios our d ive, we prese Ruge-u meod o order ree d is ierive soluio or solvig Fuzz diereil equios. Te proposed lgorim is illusred b exmple i e ls seio. Prelimir A rpezoidl uzz umber u is deied b our rel umbers k < l < m <, were e bse o e rpezoidl is e iervl [k, ] d is veries x l, x m. Trpezoidl uzz umber will be wrie s u (k, l, m, ). Te membersip uio or e rpezoidl uzz umber u (k, l, m, ) is deied s e ollowig : u(x) x l x m k k,,, k l m we will ve : () u > 0 i k > 0; () u > 0 i l > 0; () u > 0 i m > 0; d (4) u > 0 i > 0. Le us deoe R F b e lss o ll uzz subses o R (i.e. u : R [0,]) sisig e ollowig properies: (i) u R F, u is orml, i.e. x 0 R wi u(x o ) ; (ii) u R F, u is ovex uzz se (i.e. u(x ( ) ) mi{ u ( x ),u( )}, [0,], x, R); (iii) u R F, u is upper semi oiuous o R; (iv) { x R;u( x ) > 0 } is omp, were A deoes e losure o A. Te R F is lled e spe o uzz umbers (see e.g. [5]). Obviousl R R F. Here R R F is udersood s R { χ {} x ;x is usul rel umber}. x x x l m ()

3 Fuzz diereil equios 779 e deie e r-level se, x R; [ u] r { x \ u( x) r }, 0 r ; () u 0 x\ u x > 0 is omp, Clerl, [ ] { } wi is losed bouded iervl d we deoe b [u] r [ u ( r ),u( r )]. I is ler e ollowig semes re rue,. u (r) is bouded le oiuous o deresig uio over [0,],. u (r) is bouded rig oiuous o iresig uio over [0,],. u (r) u (r) or ll r (0,], or more deils see [],[]. Le D: R F R F R U{ 0 }, D ( u,v) Sup r [0,] mx { u( r ) v( r ), u( r ) v( r ) }, be Husdor dise bewee uzz umbers, were [u] r [u(r), u (r)], [v] r [ v (r), v (r)]. Te ollowig properies re well-kow (see e.g. [6]): D(u w,v w) D(u, v), u,v,w R F, D(k.u, k.v) k D(u, v), k R, u,v R F, D(u v, w e) D(u,w) D(v,e), u, v, w, e R F d (R F, D) is omplee meri spe. Lemm. I e sequee o o-egive umbers { } N sis 0 A B, 0 N -, or e give posiive oss A d B, e A A 0 B, 0 N. A Lemm. I e sequee o umbers { } N 0, {V } N 0 sis A mx {, V } B, V V A mx {, V } B, or e give posiive oss A d B, e deoig U V, 0 N, A we ve, U A U 0 B, 0 N, A were A A d B B. Lemm. Le F(, u, v) d G(, u, v) belog o C (R F ) d e pril derivives o F d G be bouded over R F. Te or rbirril ixed r, 0 r, ( 0 ) D(, ( )) L( C), were L is boud o pril derivives o F d G, d { G [ N, ( N ; r), ( N ; r) ], [ 0,] } <. C mx r

4 780 C. Durism d B. Us Teorem.4 Le F(, u, v) d G(, u, v) belog o C (R F ) d e pril derivives o F d G be bouded over R F. Te or rbirril ixed r, 0 r, e umeril soluios o ( ;r) ;r overge o e ex soluios Y (;r) d d Y (; r) uiorml i. Teorem.5 Le F(, u, v) d G(, u, v) belog o C (R F ) d e pril derivives o F d G be bouded over R F d L <. Te or rbirril ixed 0 r, e ierive umeril soluios o ( j ) ( ;r) d ( j ) ( ;r) overge o e umeril soluios ( ; r) d ( ; r) i 0 N, we j. Fuzz Iiil Vlue Problem Cosider irs-order uzz iiil vlue diereil equio is give b ( ) (, ( )), [ 0,T ] () ( 0 ) 0 were is uzz uio o, (, ) is uzz uio o e risp vrible d e uzz vrible, is e uzz derivive o d ( 0 ) 0 is rpezoidl or rpezoidl sped uzz umber. e deoe e uzz uio b [, ]. I mes e r-level se o () or [ 0, T] is we wrie [()] r [ (; r), (; r)], [( 0 )] r [ ( 0 ; r), ( 0 ; r)], r (0,] (, ) [ (, ), (, )] d (, ) F[,, ], (, ) G[,, ]. Beuse o (, ) we ve (, (); r) F[, (; r), (; r)] (4) (, (); r) G[, (; r), (; r)] (5) B usig e exesio priiple, we ve e membersip uio (, ())(s) sup{()(τ )\s (,τ )}, s R (6) so uzz umber (, ()). From is i ollows [(, ())] r [ (, (); r), (, (); r)], r (0,], (7) were (, (); r) mi { (, u) u [()] r } (8) (, (); r) mx { (, u) u [()] r }. (9)

5 Fuzz diereil equios 78 Deiiio. A uio : R R F is sid o be uzz oiuous uio, i or rbirr ixed 0 R d > 0, δ > 0 su o < δ D [(), ( 0 )] < exiss. Trougou is pper we lso osider uzz uios wi re oiuous i meri D. Te e oiui o (,();r) gurees e exisee o e deiiio o (, (); r) or [ o,t] d r [0,] [4]. Tereore, e uios G d F be deiie oo. 4 Ruge -u meod o order ree Cosider e iiil vlue problem ( ) (, ( )), [ 0,T ] (0) ( 0 ) 0 Assumig e ollowig Ruge-u meod wi ree slopes ( ) ( ) () were (, ( )) (, ( ) ) (, ( ) ) d e prmeers,,,,,, & re ose o mke.tere re eig prmeers o be deermied.now, Tlor s loser o ( ) series expsio bou gives!! ( )! ( ), ( ) [ ] [ ( )]...!!...! I we se ()

6 78 C. Durism d B. Us [ ] [ ] [ ]... ) ( ) (!,...!!, [ ] !!! Subsiuig e vlues o &, i (), we ge [ ] [ ]... ) ( () Comprig e oeiies o &, i () & (), we obi

7 Fuzz diereil equios 78,,. 6 (4) Muliplig e our d i equios b d usig e six equio o (4), we ge,. 6 6 Elimiig rom ese wo equios, we id o soluio exiss uless ( ) or. (5) 6 6 ( ) Usull,, re rbirril ose d is deermied rom (5). However, i, e we immediel obi rom e our d i equios o (4),. Te vlues o e remiig prmeers re obied rom (4). e,we ge d.e ge e vlues o e oer prmeers s 0,,, & Ruge-u meod is obied s ( ) ( ) [ ] (6) 8 ere (, ( )) d ( (,, (, ( ) ) ) ). 5 Ruge-u meod o order ree or solvig Fuzz Diereil Equios Le Y [Y,Y ] be e ex soluio d [, ] be e pproximed soluio o e uzz iiil vlue problem (). Le [Y()] r [Y ( ; r), Y ( ; r)], [()] r [ ( ; r), ( ; r)].

8 784 C. Durism d B. Us Trougou is rgume, e vlue o r is ixed. Te e ex d pproximed soluio re respeivel deoed b [Y( )] r [Y ( ; r),y ( ; r)], [( )] r [ ( ; r), ( ;r)] (0 N). Te grid pois wi e soluio is luled re T 0,i 0 i, 0 i N. N Te we obi, Y ( ; r) Y ( ; r) 8 [ ], were F[, Y ( ;r), Y ( ;r)] F[, Y ( ;r), Y ( ;r) ] (7) F[, Y ( ;r), Y ( ;r) ] d were Y ( ; r) Y ( ; r) 8 [ ], G[, Y ( ;r), Y ( ;r)] Also we ve G[, Y ( ;r), Y ( ;r) ] (8) G[, Y ( ;r), Y ( ;r) ] ( ; r) ( ; r) 8 [ ], were F[, ( ;r), ( ;r)] F[, ( ;r), ( ;r) ] (9) F[, ( ;r), ( ;r) ] d were ( ; r) ( ; r) 8 [ ], G[, ( ;r), ( ;r)] G[, ( ;r), ( ;r) ] (0) G[, ( ;r), ( ;r) ]

9 Fuzz diereil equios 785 Clerl, ( ; r) d ( ; r) overge o Y ( ; r) d Y ( ; r), respeivel weever 0. 6 Numeril Resuls I is seio, e ex soluios d pproximed soluios obied b Euler s meod d Ruge-u meod o order ree re ploed i igure d igure. Exmple 6. Cosider e iiil vlue problem () (), [ 0,] ( 0) ( r,. 0. r ). Te ex soluio is give b Y( ; r) [( r )e,(. 0.r ) e], 0 r. Usig ierive soluio o Ruge-u meod o order ree, we ve (0; r) r, (0; r). 0.r d b i ;r r i ; r (0) ( ) ( i ; ) (0) ( i ;r) ( i ; r) ( r) i ;, were i 0,,...,N d N. Now, usig ese equios s iiil guess or ollowig ierive soluios respeivel, j ( i ; r) ( i ; r) 8 [ ], were ( i ; r) ( ( i ; r) ) ( ( i ; r) ). d j ( i ; r) ( i ; r) 8 [ ], were ( i ; r) ( ( i ; r) ) ( ( i ; r) ).

10 786 C. Durism d B. Us d j,,. Tus, we ve ( i ; r) () ( r) ( i ; r) () ( i ; r), or i... N. i ; d Tereore, Y (; r) () (; r) d Y (; r) () (; r) re obied. Tble, sows esimio o error or diere vlues o r [0,] d. B miimizig e sep size, e soluio b ex meod d R meod lmos oiides. r Ex soluio , , , , , ,.788 TABLE : Ex soluio r , , , , , , , , , , , ,.784 TABLE : Approximed soluio

11 Fuzz diereil equios 787 r TABLE : Error or diere vlues o r d.

12 788 C. Durism d B. Us

13 Fuzz diereil equios 789

14 790 C. Durism d B. Us Reerees [] J. J. Bukle d E. Eslmi, Iroduio o Fuzz Logi d Fuzz Ses, Psi-Verlg, Heidelberg, Germ. 00. [] J. J. Bukle d E. Eslmi d T. Feurig, Fuzz Memis i Eoomis d Egieerig,Psi-Verlg, Heidelberg, Germ. 00. [] S. L. Cg d L. A. Zde, O Fuzz Mppig d Corol, IEEE Trs. Ssems M Cbere., (97) 0-4. [4] R. Goesel d. Voxm, Elemer Clulus, Fuzz ses d ssems, 8 (986) -4. [5] O. lev, Te Cu Problem or Fuzz Diereil Equios, Fuzz ses d ssems, 5 (990) [6] M. L. Puri d D. A. Rlesu, Diereils o Fuzz Fuios, J. M. Al. Appl., 9 (98) -5. Reeived: Jue, 009